Risk Analysis, Vol. 26, No. 1, 2006 DOI: 10.1111/j.1539-6924.2006.00705.x

Risk-Based Integrity and Inspection Modeling (RBIIM)of Process Components/System

Faisal I. Khan,1∗ Mahmoud M. Haddara,1 and Subrata K. Bhattacharya1,2

Process plants deal with hazardous (highly flammable and toxic) chemicals at extreme con-

ditions of temperature and pressure. Proper inspection and maintenance of these facilities

is paramount for the maintenance of safe and continuous operation. This article proposes a

risk-based methodology for integrity and inspection modeling (RBIIM) to ensure safe and

fault-free operation of the facility. This methodology uses a gamma distribution to model the

material degradation and a Bayesian updating method to improve the distribution based on

actual inspection results. The method deals with the two cases of perfect and imperfect inspec-

tions. The measurement error resulting from imperfect inspections is modeled as a zero-mean,

normally distributed random process. The risk is calculated using the probability of failure and

the consequence is assessed in terms of cost as a function of time. The risk function is used

to determine an optimal inspection and replacement interval. The calculated inspection and

replacement interval is subsequently used in the design of an integrity inspection plan. Two

case studies are presented: the maintenance of an autoclave and the maintenance of a pipeline

segment. For the autoclave, the interval between two successive inspections is found to be

19 years. For the pipeline, the next inspection is due after 5 years from now. Measurements

taken at inspections are used in estimating a new degradation rate that can then be used to

update the failure distribution function.

KEY WORDS: Failure modeling; integrity assessment and evaluation; quantitative risk assessment; risk-

based inspection; risk-based maintenance

1. INTRODUCTION

Maintaining the integrity of a large number ofaging process components is a subject of prime im-portance to process companies all over the world. InCanada, there are more than 250,000 km of pipelinescarrying natural gas, crude oil, and petroleum prod-ucts (Pandey, 1998). To protect the public, the fi-

1 Faculty of Engineering & Applied Science, Memorial University

of Newfoundland, St John’s, NL, A1B 3X5, Canada.2 Department of Ocean Engineering, Indian Institute of Technol-

ogy, Chennai, 600 036, India.∗ Address correspondence to Faisal I. Khan, Faculty of Engineering

& Applied Science, Memorial University of Newfoundland, St

John’s, NL, A1B 3X5, Canada; fkhan@engr.mun.ca.

nancial investment, and the environment againstthe consequences of failures occurring to these sys-tems, an assessment of the condition of existing in-frastructure (major process units, pipelines, etc.) isnecessary. The condition assessment quantifies thedegradation of the material and provides a basisfor the decision-making process regarding preventivemaintenance and/or replacement. With limited main-tenance resources, it is essential that the availablefunds be spent where they are most effective in re-ducing potential risks (Concord, 1993; Pandey, 1998).

The traditional methods of assessing the condi-tion of process systems are based on a deterministicload-resistance methodology in which the integrityof the components is evaluated by comparing the

203 0272-4332/06/0100-0203$22.00/1 C© 2006 Society for Risk Analysis

204 Khan, Haddara, and Bhattacharya

current operating conditions with a design-limit statebeyond which the component cannot operate safely.Load-resistance methods have the disadvantage thatthey often yield somewhat conservative results, lead-ing to potentially unnecessary repairs and inspectionsthat result in an overall increase in maintenance costs.Use of deterministic methods does not provide in-formation about potential risk that results in theunrealistic maintenance planning for process plants(Desjardins, 2002).

Risk is defined as multiplication of the probabil-ity of failure and its likely consequence. Risk-basedmethods aim at identifying, characterizing, quantify-ing, and evaluating the likelihood of the loss causedas a result of the occurrence of a specific event. Theuse of risk-based methods for the management of theprocess components provides reliable quantificationof potential risks. This provides an alternative strat-egy for the maintenance of assets instead of the use ofsimple ranking (prioritizing) based on reported fail-ure occurrences.

This approach also provides a means for quan-titatively establishing future reliability levels for thecomponents. These levels can be used as a basis foroptimizing reinspection intervals. The uncertaintiesassociated with the design and operation of processcomponents have led to an increasing use of risk-based approaches in making decisions regarding assetintegrity management.

In risk-based inspection (RBI) strategy risk isused as a criterion to prioritize inspection tasks for thecomponents in a process plant. This provides manyadvantages, which include (1) an increase in plantavailability, (2) a decrease in the number of failureoccurrences, (3) a reduction in the level of risk dueto failure, and (4) a reduction in the direct inspec-tion cost of the plant. Operational safety of a pro-cess plant increases as a result of decreasing the num-ber of failure occurrences and the potential risk dueto failure. This makes RBI a useful strategy to meetthe rising societal expectations regarding operationalsafety of complex onshore and offshore process fa-cilities. These expectations resulted in a number ofregulations that the oil and gas industry has to meet.The last two decades saw a number of studies that at-tempted to address the subject. Different methodolo-gies were suggested to meet maintenance planning re-quirements. These methodologies range from the fullyqualitative to the fully quantitative. A brief summaryof these approaches is presented in the subsequentsection.

2. AVAILABLE RISK-BASED APPROACHES

Since the late 1980s, numerous quantitative, semi-quantitative, and qualitative models have been devel-oped to aid plant engineers with the prioritization ofcomponents inspections. In the following sections, weoutline briefly these methods.

2.1. Qualitative Approaches

Qualitative risk index approaches assign subjec-tive scores to the different factors that are thought toinfluence the probability and consequences of failure(Muhlbauer, 1992; Cagno et al., 2000; Dey & Gupta,2001). These scores are then combined using simpleformulas to give an index representing the level ofrisk. Risk index approaches provide a ranking of thedifferent process components based on the perceivedlevel of risk estimated. The ranking obtained by us-ing these methods is highly subjective. In addition,these approaches do not provide any indication ofwhether the risk associated with a component is un-acceptable and consequently no guidance is providedregarding whether any risk reduction action is nec-essary. The index system scoring format suggestedby Muhlbauer (1992) accounts for the use of in-lineinspection tools to locate metal loss corrosion byawarding up to 8 points out of a maximum of400 points representing resistance to failure (i.e., 2%).This underestimates the benefits of high-resolutionpigging, which is known to result in significant re-ductions to the large percentage of failures that areattributable to corrosion (20–40% of all failures).Therefore, index systems provide at best an ap-proximate risk-based ranking of process components,which has serious limitations when being used as abasis for integrity management decision making.

2.2. Quantitative Approaches

These approaches determine the level of riskbased on direct estimates of the probability and/orconsequences of failure. Current quantitative risk as-sessment approaches focus on a single aspect of theconsequence associated with failure. Published stud-ies deal with either loss of life risk or economic risk(Hill, 1992; Concord, 1993; Nessim & Stephens, 1995;Pandey, 1998; Nessim et al., 2000).

Integration of environmental damage, life safety,and economic risks has not been addressed ad-equately. Another limitation of quantitative risk

Risk-Based Integrity and Inspection Modeling 205

assessment approaches is that they typically base thefailure probability estimates on historical failure rates.Publicly available databases do not usually allow sub-division of the failure data according to the attributesof a specific process component and where adequatesubdivision is possible, the amount of data associ-ated with a particular attribute set is very limited be-cause of the rarity of the failures. Failure probabilitiesestimated from public data are, therefore, not suffi-ciently specific to represent a given failure in a specificprocess component.

Another approach to calculate failure probabil-ity is based on the concept of structural reliabilitythat includes Markovian models and hot spots on acomponent-by-component basis (a number of inter-esting articles may be seen in Journal of Infrastruc-ture Systems). The effect of the correlation from onehot spot to another has been investigated by Lotsberget al. (1998), Faber and Sorensen (1999), Brown andMay (2000), and Montgomery and Serratella (2002).Faber et al. (2000, 2003) proposed an informal decisionanalysis where the number of considered elements isreduced in a consistent and systematic way. However,due to the numerical effort required and the stabil-ity of the method such an approach did not prove tobe practical. Straub and Faber (2000) indicated thatan integrated approach to the decision problem thatis suitable for industrial purposes has not yet beendeveloped.

2.3. Semi-Quantitative Approaches

Semi-quantitative approaches were developed toprovide a practical and easy tool to be used for de-signing maintenance programs that optimize the useof resources and in the meantime ensure effective andefficient asset management. These approaches usesemi-quantitative models for consequence estimationas well as failure probability calculations. Examples ofthese approaches can be found in Khan and Haddara(2003a, 2003b), Khan and Haddara (2004), and Khanet al. (2004). It is easily employed in process plants andto components like pipelines or pressure components.These approaches provide a tool to ascertain that theestimated risk of failure satisfies a predetermined ac-ceptance criterion (Khan et al., 2004; Willcocks & Bai,2000; Dey, 2004).

The risk-based inspection and maintenance ap-proach discussed by Willcocks and Bai (2003) usesa failure modes effect analysis to identify the failuremodes of system components and their consequences.It then uses failure patterns and rates to calculate the

probability of failure, and it determines the risk tobe used in inspection and maintenance planning. De-pending on the level of risk for each mode and patternof failure, the required analysis, inspection, mainte-nance, and repair tasks are selected. For example, areview of historical failure databases indicates thatthe major failure modes in a pipeline are internal cor-rosion and external impact. Thus, the main efforts (interms of design, structural modeling, inspections, etc.)should be focused on these failure modes. Of course,this is a simple example of risk-based inspection andmaintenance. In practice, more specific details aboutthe specific pipeline need to be considered.

Guidelines to help hydrocarbon and processchemical industries in establishing risk-based inspec-tion programs on fixed equipment and piping wereissued by the American Petroleum Institute (API,2002). This document describes a recommendedpractice for developing risk-based maintenance pro-grams. The document does not preclude the use of anyof the three methodologies mentioned above, but out-lines the limitations of each methodology. This docu-ment explains the basic elements for developing andimplementing a risk-based inspection program.

2.4. Maintenance Optimization

A large number of articles were published on thesubject of optimizing maintenance through the useof mathematical models (Montgomery & Serratella,2000; Khan & Haddara, 2003a, 2003b; Willcocks &Bai, 2003; Dey, 2004). Most maintenance optimiza-tion models are based on lifetime distributions orMarkovian deterioration models. It is often difficult tocollect enough data for estimating the parameters ofa lifetime distribution or the transition probabilitiesof a Markov chain. This presents an obstacle in theway of using these models to design practical mainte-nance programs. The combined use of the reliabilityindex methods and the limit state approach may provehelpful in removing this drawback.

Typically, structural reliability methods are usedto estimate the probability of failure of componentsor structures. In these methods, the condition of thematerial is described by a state function g. The statefunction represents the strength margin that the ma-terial has over the applied load. Thus, the limit-statefunction, gt is defined as

gt = R − L, (1)

where R is the strength of the material and L is theapplied stress.

206 Khan, Haddara, and Bhattacharya

As degradation of material occurs and operat-ing conditions change, material strength and appliedstress become random functions of time. Thus, thelimit-state function becomes a stochastic process. Afailure model can be developed based on the limit-state function approach and allowing for materialdegradation variability, operating conditions variabil-ity, and measurement errors. Data collected during in-spections provide a means to update the failure modelthrough the use of Bayes’s theorem.

Risk-based maintenance takes into considerationthe consequence of failure as well as the probabilityof failure. In this article, the consequence is measuredin terms of the cost of failure in addition to the costof the inspection/maintenance program.

3. QUANTITATIVE RISK-BASED INTEGRITYAND INSPECTION MODELING (RBIIM)

RBIIM aims at modeling inspection tasks toachieve safe operating conditions at minimum cost.Determining the relationship between inspectionsperiodicity on one side and the level of safety andmaintenance costs on the other, provides an insightinto the application of the as low as reasonable prac-ticable (ALARP) principle. In order to implementan integrated RBIIM approach, it is necessary firstto formulate acceptance criteria for each functionof the plant facility. Thereafter, the relationship be-tween each function and the performance of thevarious components responsible for achieving thisfunction is established. To simplify the implemen-tation, the acceptance criteria for each function ofthe facility are transformed into acceptance criteriafor the individual components. Subsequently, risk as-sessment is performed relating the event of compo-nent failure to the consequences in terms of monetarylosses. The main objectives of RBIIM are:

1. To identify critical equipment in a processplant. These are pieces of equipment that havea high level of risk.

2. To develop an optimum inspection and main-tenance strategy for the equipment that willguarantee the integrity of the plant.

Fig. 1 depicts the overall framework designed toachieve these objectives. The framework comprisesfive stages. These are: identification of equipmentto be analyzed, identification of degradation mech-anisms for each component, calculation of the riskassociated with the failure of each component, deter-

mination of an optimal inspection policy (interval) foreach component, and development of a comprehen-sive policy for the plant integrity assessment.

3.1. Material Degradation Mechanisms

Material degradation is a common cause of pro-cess components failure. Material degradation can becaused by one or more mechanisms. These include:internal and external thinning due to corrosion; stresscorrosion cracking; brittle fracture; and fatigue due tovibration. These mechanisms cause material deterio-ration and thus affect the ability of the component towithstand the applied load.

Two basic models will be used in this work todescribe material degradation mechanisms: a thin-ning model and a crack model. The thinning modelis used to describe the reduction in the thickness ofthe component’s material as a result of internal orexternal corrosion. The crack model is used to de-scribe the reduction of the load-carrying capacity ofthe component as a result of cracks. Cracks may resultfrom stress corrosion, brittle fracture, or fatigue (seeKallen, 2002).

3.1.1. Thinning Model

Corrosion wear failure is a complex failure modein which a component is affected by a combination ofcorrosion and wear. Continuous wear results in mate-rial thinning, which may be localized or can spreadover a large surface area. In the former, materialdegradation is seen in the form of pitting, while in thelatter, a general thickness loss occurs. Carbon steel orcopper are usually more susceptible to general thin-ning, while stainless steels and higher alloy materi-als are usually more prone to localized thinning andpitting.

The state function in this case, gt, is defined as

gt = R − St = S(

1 − C × �td

)−

(P × D2 × d

), (2)

where R is material resistance, St is the applied stress,S is material strength, C is corrosion rate, P is operat-ing pressure, D is the diameter of the component, d isthe material thickness, and �t is a time increment.

Corrosion can occur on the internal or externalsurface of the component. The same thinning modelcan be used for both cases; however, the corrosionrate may differ from one case to the other.

The state function gt, is a measure of the abilityof the material to resist failure. Failure will occur onlyif the magnitude of the function gt reaches zero.

Risk-Based Integrity and Inspection Modeling 207

Divide the system into independent (process) components, e.g., vessel, pipeline

segments, reactors, etc.

Select one component at a time and identify possible degradation mechanisms

Model component failure using a gamma stochastic process

Estimate the consequence using cost data for inspection, failure, and replacement

Develop a prior for each degradation mechanism

Determine failure probability using a posterior and past inspection results

Risk calculations

Estimate optimal inspection and replacement intervals

Have all components been considered?

Develop inspection plan for the integrity of the process system

Obtain past inspection results for the unit under investigation

Fig. 1. Framework of risk-based integrity

and inspection modeling methodology.

Equation (2) can be used to define a margin ofsafety in terms of material thickness. Assuming a lin-ear degradation model, the incremental wall thicknessloss (�x) during an element of time �t, is given by

�x = C�t =(

1 − PD2dS

)d. (3)

The margin of safety in terms of the minimum re-quired material thickness, mt, is then obtained as

mt = d − PD2S

. (4)

3.1.2. Stress Corrosion Cracking Model

Stress corrosion cracking (SCC) failure occurswhen the applied stresses on a component in a corro-sive environment generate a field of localized surfacecracks, usually along grain boundaries, which rendersthe component incapable of performing its function.

208 Khan, Haddara, and Bhattacharya

There are many material/environmental combina-tions within which SCC can occur. They include caus-tic cracking, amine cracking, carbonate cracking, sul-fide stress cracking, hydrogen-induced cracking, andpolythionic acid cracking, and chloride cracking forstainless steels (Kallen, 2002).

The SCC state function uses a resistance minusstress model based on Paris’s crack growth law usedin linear elastic fracture mechanics,

gc = KIC − Y(

PD2d

+ S) √

π A, (5)

where KIC is the material fracture toughness, Y is adimensionless geometric factor, S is residual stress,and A is the crack depth.

The crack depth, A, is determined by

lcr = Ccr�tn,

A = lcr

Rl/a,

(6)

where lcr, Ccr, and Rl/a are the characteristic cracklength, the crack growth rate, and the crack to length-to-depth (thickness) ratio, respectively. The valueof KIC, material fracture toughness, is taken to be300 ksi(in)1/2 (10,425 MPa (mm)1/2) for stainless steel.The material fracture toughness for carbon and lowalloy steel may be calculated using the following for-mula (Kallen, 2002):

KIC

= Minimum{33.2 + 2.806e(0.02(T+100)), 200

}ksi

√in.

Equation (6) can be used to derive an expression forthe incremental increase in crack depth as

�x = Ccr�t = Rl/a

π

⎡⎢⎢⎣ KIC

Y[

PD2d

+ S]⎤⎥⎥⎦

2

. (7)

The safety margin for wall thickness, mc, becomes

mc = d − Ccr�t. (8)

3.1.3. Brittle Fracture and Fatigue

At low temperatures carbon steels may suffera brittle fracture at loads significantly below the in-tended design loads. This is because carbon steels losetheir ductility as the temperature is lowered. The tem-perature at which ductility is lost is called the transienttemperature. While most materials are designed tooperate well above the transition temperature, somematerials in older plants may still operate near or

below the transition temperature and as a result aresusceptible to brittle fracture. Furthermore, vibrationmay cause components to fail prematurely. Improp-erly supported piping near vibration sources is proneto fatigue. In contrast to the other degradation mech-anisms discussed above, this mechanism is modeledusing a simple indexing methodology (Kallen, 2002).

3.2. Material Degradation Model

In the previous section, we reviewed the differ-ent mechanisms that control material degradation.The net degradation of the material is the total sumof degradations that take place as a result of one ormore of the above-mentioned mechanisms. Althoughthe above-mentioned mechanisms are deterministic,there is a level of uncertainty associated with some oftheir variables. For instance, the strength S, the cor-rosion rate C, and the operating pressure, P in thethinning model, and KIC, Y, P, S, and A in the stresscorrosion cracking mechanism, have uncertainty as-sociated with their values. Therefore, these variableshave to be considered random and the material degra-dation process is expected to be a stochastic process.

Let us define the accumulated material degrada-tion measured at a certain point in time t, to be thesum of the incremental degradations that occurredfrom the start of service till time t. Let us also as-sume that the incremental material degradations areindependent, exponentially distributed random vari-ables. Then, the cumulative degradation from the startof service till time t, X(t), is a gamma-distributedstochastic process with stationary increments. Thegamma-distributed stochastic process has many ad-vantages: it has a nonnegative distribution and it iseasy to manipulate.

A gamma density function with a shape parame-ter α, and a scale parameter β, where α and β > 0, isgiven by

f (x) = βα

�(α)(x)α−1 e−βx for x > 0. (9)

For the thinning model, the cumulative degradation isassumed to be linear in time. Thus, the shape param-eter, α, will be replaced by a linear function of timeα = αo t .

The probability density function describing thecumulative wall thickness degradation, X(t), is de-scribed by Equation (10):

Risk-Based Integrity and Inspection Modeling 209

fX(t)(x) = βαot

�(αot)(x)αot−1e−βx for x > 0. (10)

This distribution will be used to describe cumulativematerial degradation caused by each individual degra-dation mechanism described above.

Thus, the mean and variance of the cumulativedegradation in the wall thickness, X(t), are given by

μX(t) = E[X(t)] = α0

βt = μt,

σ 2X(t) = Var[X(t)] = αo

β2t2 = σ 2t. (11)

The parameters of the gamma process, X(t), are cal-culated as

αo =(μ

σ

)2

and β = μ

σ 2. (12)

It is common practice to express the standard devia-tion (σ ) in terms of the mean (μ) through the use ofa coefficient of variance (υ): σ = υ μ. Using this re-lationship, the gamma density function for corrosioncan be rewritten as

fX(t)(x) =

(1

μν2

) tν2

�

(tν2

) (x)t

ν2 −1 e− xν2μ . (13)

In this expression the average corrosion rate (μ) is theonly uncertain parameter that would vary with time.Considering this, the above expression can be repre-sented in terms of single-variable Bayesian gammaconditional function G as

G[

x

∣∣∣∣ 1

ν2t,

1

μν2

]

=

(1

μν2

) tν2

�

(tν2

) (x)t

ν2 −1 e− xν2μ = fX(t)(x). (14)

For stress corrosion cracking, the cumulative crackdepth is assumed to be a function of tn. Thus, in thiscase α = αotn and the density function becomes

fX(t)(x) = βαot

�(αotn

) (x)αotn−1e−βx for x > 0. (15)

The mean and variance for the cumulative wall thick-ness degradation become

μX(t) = E[X(t)] = αo

βtn = μtn,

σ 2X(t) = Var[X(t)] = αo

β2tn = σ 2tn. (16)

The parameters of the gamma process, X(t), are cal-culated as

αo =(μ

σ

)2

and β = μ

σ 2. (17)

Substituting for the parameters of the density func-tion, one gets

fX(t)(x) =

(1

μν2

) tn

ν2

�

(tn

ν2

) (x)tn

ν2 −1 e− xν2μ

= Ga[

x

∣∣∣∣ 1

ν2tn,

1

μν2

]. (18)

Similarly, the above equation can be represented interms of a single variable conditional gamma functionG as

G[

x

∣∣∣∣ 1

ν2tn,

1

μν2

]

= fX(t)(x) =

(1

μν2

) tn

ν2

�

(tn

ν2

) (x)tn

ν2 −1 e− xν2μ . (19)

These gamma density functions are subsequently usedin the inspection updating for the cases of perfect andimperfect inspections.

3.3. Inspection Updating Modeling

The results of inspection can be effectively used inupdating the prior knowledge of the average degrada-tion rate (μ). This updating could be best done usingBayesian updating. Bayes’s theorem provides a for-mal and structured approach that can be used to up-date the prior based on component data gained fromactual inspections. Inspection updating modeling in-volves two main steps: (1) selection of an appropriateprior, and (2) Bayesian updating of the prior usingnew inspection data. Bayesian updating can be ap-plied for the two cases: perfect or imperfect inspectiondata. Bayesian updating is widely discussed in the lit-erature; excellent reviews are given by Kallen (2002)and Kallen and Noortwijk (2003).

3.3.1. Step I—Selection of a Prior

Bayes’s theorem indicates that a posterior proba-bility can be obtained from an assumed prior prob-ability when modified based on new knowledge

210 Khan, Haddara, and Bhattacharya

(information). A number of probability distributionshave the special property that if used in Bayes’s the-orem as a prior distribution and the new informa-tion is used in the form of a random variable witha certain distribution, the posterior distribution hasthe same distribution as the prior. These pairs ofdistributions are called conjugate distributions, ex-amples of which are normal-normal-normal, normal-lognormal-normal, and gamma-exponential-gammadistributions (see Harr, 1987).

Following the suggestions made by van Noortwijkand van Gelder (1996), Kallen (2002), and Kallen andvan Noortwijk (2003), the inverted gamma distribu-tion is used here as a prior distribution. The invertedgamma density is given as

IX(t)(x) = ba

�(a)

(1

x

)a+1

exp

(−b

x

)for x ≥ 0.

(20)

An inverted gamma density fits very well as a priordensity function for the material degradation causedby corrosion and stress corrosion cracking. The den-sity is nonnegative and has a longer tail, which repre-sents uncertainty over higher degradation rates. Fig. 2depicts the simple discrete prior density and also thecontinuous version of this density function. The in-verted gamma density function has been fitted tothe discrete prior given in Fig. 2, through the use ofsimulation.

0

10

20

30

40

50

60

70

80

0.00 0.06 0.11 0.17 0.22 0.28 0.34 0.39 0.45 0.50 0.56 0.60 0.70

Degradation rate (mm/year)

Den

sit

y (

%ag

e)

Discrete prior (based on expert opinion)

Inverted gama prior

Fig. 2. Discrete prior based on available

information and inverted gamma prior

fitted to these data.

3.3.2. Step II(i)—Posterior: PerfectInspection Updating

The conditional prior density function for the de-crease in the thickness of the material, X(t), can beexpressed as

π(μ | x) = l(x | μ)π(μ)∫ ∞

0

l(x | μ)π(μ) dμ

, (21)

where π(μ) is the prior density for μ, and l(x | μ)is the likelihood of a measurement x given μ.

Kallen (2002) used Equation (21) to obtain aposterior density function for the loss in the mate-rial thickness for the perfect inspection case, π(μ | x),given by

π(μ | x) = IX(t)

(μ

∣∣∣∣ tυ2

+ α,xυ2

+ β

), (22)

where α and β are the parameters of the prior density.Consider the case when the inspection is repeated

n times, thus, n measurements x1 at time t1, x2 att2, . . . , xn at tn are carried out. The posterior for theloss in the material thickness can be written as

π (μ | x1, . . . , xn)

= IX(t)

(μ

∣∣∣∣∣ n∑i=1

ti − ti−1

υ2+ α,

n∑i=1

xi − xi−1

υ2+ β

).

(23)

Risk-Based Integrity and Inspection Modeling 211

If one considers that xo = 0 at to = 0, then the posterioractually depends only on the last inspection, becausethe standard deviation (σ ) is expressed in terms of themean (μ). Therefore, the above expression reduces to

π(μ | x1, . . . , xn)

= π(μ | xn) = IX(t)

(μ

∣∣∣∣∣ n∑i=1

tnυ2

+ α,

n∑i=1

xn

υ2+ β

).

(24)

Perfect inspections do not usually happen in real life;it is thus important to model imperfect inspections aswell.

3.3.3. Step II(ii)—Posterior: ImperfectInspection Model

This model is based on the approach discussedby Newby and Dagg (2002), and Kallen and vanNoortwijk (2003). Let us consider a stochastic pro-cess Y(t) given by

Y(t) = X(t) + ε, (25)

where ε is the error in the measurement. The error, ε,is considered to be a zero mean normal process witha standard deviation σ ε. Y(t) is the degradation mea-sured (material lost) during an inspection at time t.

A brief description of the posterior density for-mulation and solution algorithm is presented here;for more details see, Kallen (2002) and Kallen andvan Noortwijk (2003). The likelihood of a measure-ment y given the degradation rate μ is

l(y | μ) = fY(t)(y)

= fX(t)+ε(x) =∫ ∞

−∞fX(t)(y − ε) fε(ε) dε,

(26)

where f X (t)(y − ε) is the likelihood of the gamma in-crement, X(t), having parameters α and β. To solvethe above integral, we replace the integral in Equa-tion (26) by a summation

fY(t)(y) ≈n∑

i=1

G(y − εi

∣∣ αotn, β)

p(εi );

where �p(εi ) = 1. (27)

Equation (27) was developed for a nonlinear materialdegradation model: a linear model can be obtained bysubstituting n = 1.

Using the parameters of the gamma density func-tion, the denominator in Equation (21) becomes

∫ ∞

0

l(y | μ)π(μ) dμ

=∫ ∞

0

{∫ ∞

ε=0

G(

y − ε

∣∣∣∣ 1

υ2tn,

1

μυ2

)fε(ε) dε

}π(μ) dμ.

(28)

This integral can also be approximated by the follow-ing summation:∫ ∞

0

l(y | μ)π(μ) dμ

≈n∑

i=1

{k∑

j=1

G(

y − ε

∣∣∣∣ 1

υ2tn,

1

μυ2

)p(ε j )

}p(μi ),

(29)

where p(μi ) = Pr{μ = μi} is the discrete density ofμ. Using these results, the posterior is approximatedas

p(μi | y)

=p(μi )

{k∑

j=1

G(

y − ε

∣∣∣∣ 1

υ2tn,

1

μυ2

)p(ε j )

}n∑

i=1

p(μi )

{k∑

j=1

G(

y − ε

∣∣∣∣ 1

υ2tn,

1

μυ2

)p(ε j )

} ;

for i = 1, . . . , n. (30)

In case of multiple imperfect inspections, we need theproduct of the individual likelihoods for the measuredincrements

l(y1, . . . , yk | μ) =∏

k

lY(tk)−Y(tk−1)(yk − yk−1 | μ).

(31)

Using the convolution integral as before, the abovelikelihood can be rewritten as

l(y1, . . . , yk | μ)

=∫ ∞

−∞. . .

∫ ∞

−∞

∏k

fX(tk)−X(tk−1)(yk − yk−1 − δk)

× f (δ1, . . . , δk) dδ1, . . . , dδk, (32)

where δk = εk − εk−1. Because the δs are not indepen-dent, the likelihood can be approximated as

l(y1, . . . , yk | μ) = E

[∏k

fDk(dk − δk)

]

≈ 1

N

N∑j=1

fDk(dk − δ

jk

)as N → ∞, (33)

where Dk = X(tk) − X(tk−1) and dk = yk − yk−1. Inthe last step we used the strong law of large numbers

212 Khan, Haddara, and Bhattacharya

to approximate the expectation of a function throughthe use of simulation (see Kallen, 2002).

For each k, we sample εk (j = 1, . . . , N) and calcu-

late δjk = ε

jk − ε

jk−1. Because the gamma distribution

f Dk = G(x | α[tk − tk−1], β) is not defined for x < 0,

we need to make sure that the condition (dk − δjk) ≥

0 is satisfied.

l(y1, . . . , yk | μ)

≈ 1

N

N∑j=1

∏k

Ga(

dk − min{δ

jk, dk

}∣∣∣∣ tk − tk−1

υ2,

1

μυ2

).

(34)

The posterior for multiple imperfect inspections cannow be obtained as

p(μi | y)

=p(μi )

1

N

N∑j=1

∏k

Ga

(dk − min

{δ

jk, dk

} ∣∣∣∣∣ �tυ2

,1

μυ2

)n∑

i=1

p(μi )1

N

N∑j=1

∏k

Ga

(dk − min

{δ

jk, dk

} ∣∣∣∣∣ �tυ2

,1

μυ2

) ,

(35)

for i = 1, . . . , n and �tk = tk − tk−1.We now have a Bayesian updating model that ac-

cepts the results of multiple imperfect inspections. Theadvantage of using the simulation technique to calcu-late the posterior density as a function of the degra-dation rate is that one can use different levels for themagnitude of the error in each inspection.

3.4. Risk-Based Optimal InspectionInterval Calculation

Determining an optimal integrity inspectionand/or maintenance action is a problem of optimiza-tion under uncertainty. The most comprehensive ap-proach for solving such a problem is through the useof decision theory, which provides a systematic andconsistent way to evaluate the alternatives and iden-tify an optimal choice (see Nessim & Stephens, 1995;Nessim et al., 2000; Goyet et al., 2002).

In the previous steps, we have developed a modelto calculate the probability of failure of a process com-ponent as a function of time. In order to calculate therisk, we have to calculate the consequences associatedwith that failure. In the present study, we estimate theconsequence in terms of the cost incurred as a resultof failure. Having estimated the risk, we will use itto decide when to inspect and when to replace thecomponent. In other words, we intend to determinethe maximum length of time between two consecutiveinspections that would result in a minimum accepted

risk. We will adopt an optimization criterion that hasbeen suggested by Wagner (1975). This criterion is re-ferred to as “the average cost over unbound region.”This criterion is derived using renewal theory and usesthe concept of a component’s lifecycle. The length ofthis cycle spans the time from the start of the serviceuntil a renewal is affected. Renewal consists of eithera preventive or a corrective replacement. The risk fac-tor is calculated as the ratio of the expected costs percycle over the expected cycle length. A brief formu-lation of this criterion is presented below (for details,see van Noortwijk, 2000, 2003).

We will model the maintenance tasks as a dis-crete renewal process. In this case, the componentreturns to the as-good-as-new condition after eachreplacement. A discrete renewal process {N(n), n =1, 2, 3, . . . } is a nonnegative integer-valued stochas-tic process that registers the successive renewals inthe time interval (0, n). Let the renewal times T1, T2,T3, . . . , be nonnegative, independent, identically dis-tributed, random quantities having a discrete proba-bility function

Pr {Tk = i} = pi , where i = 1, 2, . . . ,

and∞∑

i=1

pi = 1, (36)

where pi represents the probability of a renewal attime i. The cost (consequence) associated with a re-newal at time i is denoted by ci, i = 1, 2, 3, . . . . Therisk factors (expected average costs per unit time)are determined by simply averaging the risk factorover an unbounded horizon. The expected risk over abounded horizon (0, n), denoted by R(φ, τ ), is givenby

R(φ, τ ) =n∑

i=1

pi{ci + E(τ (n − i))

},

for n = 1, 2, 3, . . . , and τ (0) ≡ 0. (37)

The cost associated with the occurrence of the eventTi = i is ci plus the additional expected cost during theinterval (i, n), i = 1, . . . , n. Using the discrete renewaltheorem, the risk factor (expected average costs perunit time) is expressed as

R(φ, τ ) = E(cycle risk)

E(cycle length)

= E(Ci )

E(I)=

∞∑i=1

ci (φ, τ )pi (φ, τ )

∞∑i=1

ipi (φ, τ )

, (38)

Risk-Based Integrity and Inspection Modeling 213

where ci and pi are the cost incurred during a unitof time i and the probability of renewal during thistime, respectively. φ{m, ρ} represents the replace-ment level, which is the ratio of the material degrada-tion to the critical safety margin. It is calculated foreach time interval τ . The random variable I is the cy-cle length with a cycle cost of cI . Considering replace-ment cycle as the time period between two renewals,the numerator of Equation (38) represents expectedcycle risk and the denominator is the expected cyclelength (mean lifetime). Equation (38) can be rewrit-ten as

R(φ, τ )

=

n∑j=1

{ ∞∑i=1

ci (φ, τ ) pi (φ, τ, μ | P, S . . .)

}p (μ j | y)

n∑j=1

{ ∞∑i=1

ipi (φ, τ, μ | P, S . . .)

}p (μ j | y)

,

(39)

where π(μ | y) is the posterior density, (p(μj | y)) is thediscrete version of the posterior density over a givennumber of the measurements, y = y1, y2, . . . , yk.

In order to include the uncertainty over the var-ious material and operating characteristics such aspressure (P) and material strength (S), we can sam-ple these variables from their respective probabilitydistribution functions. It is better to include the un-certainty over the degradation rate in the simulation,instead of using a discrete solution. The advantage ofusing a simulation technique is that the degradationrate, μ, can be sampled before we go into the loops tocalculate the expected average risk. Also, there will beno extra loop needed if we assume that other param-eters such as the pressure, P, and material strength,S, are also uncertain. The solution in this case is givenby

E|R(φ, τ )|

=

1

N

N∑j=1

{ ∞∑i=1

ci (φ, τ ) pi(φ, τ | μ j , P j , Sj . . .

)}IN

n∑j=1

{ ∞∑i=1

ipi(φ, τ | μ j , P j , Sj . . .

)} ,

(40)where E|R(φ, τ )| denotes expected average risk andis measured as dollar per unit time.

4. CASE STUDIES

4.1. An Autoclave

Geary (2002) presented a review of risk-basedinspection approaches practiced in the United King-dom. Four different case studies were performed byseven different agencies. In the present article, wewill apply the methodology outlined above to thecase study no. 3 of Geary’s report: an autoclave. Wewill determine the optimal inspection interval for thiscomponent. A brief description of the autoclave ispresented here and the input data used in the study isgiven in Table I.

4.1.1. Process Details

The autoclave reactor is filled with water at 75◦Cand pressurized to 10 Barg. The pressure is subse-quently released and water is then agitated and a vac-uum is applied to evacuate air from the empty space.Subsequently, an ammonium sulfate solution (as acatalyst) is introduced along with tetrafluoroethylene(TFE) (in a gaseous state). TFE is immediately poly-merized by the water. The temperature of the exother-mic reaction is controlled by cooling the water jacket(circulating water at 15◦C). The process operates inbatches, 3 cycles per day for 48 weeks in a year.

The vessel has been inspected three times and theresults of the inspections are given in Table I. This unithas been studied using the methodology outlined inthe previous sections and results are briefly discussedbelow.

5. RESULTS AND DISCUSSIONS

The prior and posterior density functions for thin-ning and cracking in the autoclave’s material areshown in Figs. 3 and 4, respectively. It may be observedfrom Fig. 3 that the posterior supports the prior. Theposterior for perfect inspections shows a tall peak ata corrosion rate of 0.185 mm/year, which is compara-tively higher than the prior peak. Peaks for the pos-terior (imperfect inspections) and the prior occur atalmost the same location (0.11 mm/year). This indi-cates that the results of the inspection (with possibleerrors in the measurements) confirm the suitability ofthe originally assumed thinning rate.

In case of crack growth, the peak of the poste-rior (perfect inspections) function occurs at a degra-dation rate of 0.2 mm/year, which is much higher thanthat for the prior given as 0.012 mm/year and theposterior for the imperfect inspections occurring at

214 Khan, Haddara, and Bhattacharya

Table I. Input Details for Autoclave

Background information Vessel manufactured in 1980

Design temperature 100◦C and pressure 25 Barg

Operating temperature 8–95◦C and pressure 10 Barg

Autoclave dimensions 5 ft diameter; 11′diameter, 1 in. wall thickness

Material of construction Stainless steel 321, carbon steel jacket

Active damage mechanism Stress corrosion cracking

Internal thinning

Inspection history Three inspections

Material loss (thinning): mean 0.7 mm 0.9 mm 1.5mm,

CoV 0.50

Crack growth: mean: 0.06 mm 0.3 mm 0.6 mm, CoV 0.20

Other Parameters ValuesTensile strength Mean 515 MPa, coefficient of variance (CoV) 0.20

Operating pressure Mean 3.2 MPa, CoV 0.05

Corrosion rate Mean 0.1 mm/year; CoV 0.25

Crack growth rate Mean 0.01 mm/year; CoV 0.25

Corrosion allowance 4.5 mm

n (power function for crack 0.50

growth rate)

Crack length to depth ratio 6

Inspection cost $15,000

Preventive replacement cost $30,000

Failure cost $3,000,000

a rate of 0.04 mm/year. This indicates that the poste-rior function for the perfect inspections represents anidealized condition and does not represent the actualdegradation mechanism in this case. Furthermore, theprior and posterior functions for the imperfect in-spections are fairly close together, which indicatesthat they give a better representation of the actualsituation.

0

5

10

15

20

25

30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Corrosion rate (mm/year)

De

ns

ity

prior

postprior

post_imp_prior

post_imp_prior_CDF

Fig. 3. Prior and posterior density

distribution for corrosion degradation

mechanism.

Calculations for the life cost were made using theimperfect inspections assumption and considering allpossible variations in the operating parameters. Theresults are shown in Figs. 5 and 6 for the thinning andcracking mechanisms, respectively. It can be seen fromFig. 5 that the replacement time is 43 years and that thefailure time is about 212 years. There are two minimaobserved in the plot. The first is at 43 years (which

Risk-Based Integrity and Inspection Modeling 215

0

10

20

30

40

50

60

70

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Crack growth rate (mm/year)

De

ns

ity

0

0.2

0.4

0.6

0.8

1

1.2

Cu

mm

ula

tiv

e d

en

sit

y f

un

cti

on

prior

postprior

post_imp_prior

post_imp_prior_CDF

Fig. 4. Prior and posterior density

distribution for crack growth.

coincides with the replacement interval length) andthe second is at 203 years. The second minimum isa mathematical result that has little meaning in reallife. The first minimum, which occurs at 43 years, isthe one of interest. Considering that the componenthas already survived for 24 years, the next inspectionshould be within 19 years.

It is evident from Fig. 6 (crack growth) that pre-ventive replacement time is estimated at 91 years

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 5 0 100 150 200 250

Inspection interval (years)

Avera

ge c

ycle

co

st

($/y

ear)

Average cycle cost

Replacement time

Failure timeFig. 5. Average inspection cost/year for

corrosion degradation mechanism.

whereas the failure time is calculated to be 554 years.This plot has also two minima: one at 85 years and theother at 535 years. The second minimum is imprac-tical. Now, as the autoclave has already survived for24 years, the next inspection should be done withinthe next 50 years, assuming SCC to be the only degra-dation mechanism. These results were obtained byconsidering the effects of only one degradation mech-anism at a time. This is partially responsible for the

216 Khan, Haddara, and Bhattacharya

0

1000

2000

3000

4000

5000

6000

0 100 200 300 400 500 600

Inspection interval (years)

Avera

ge c

ycle

co

st

($/y

ear)

Average cycle cost

Replacement time

Failure time Fig. 6. Average cycle cost ($/year) for

cracking degradation mechanism.

rather long inspection interval. The second reason forthe long inspection interval is the fact that we haveused a relatively slow crack growth rate.

Considering both degradation mechanisms to beacting at the same time, it is evident that the autoclaveshould be inspected at a maximum of 19 years fromnow. The analysis suggested here can be used repeat-edly to modify the inspection interval based on newerinspection data.

6. A PIPELINE

Pandey (1998) studied the probabilistic modelsfor the assessment of the condition of a pipeline us-ing in-line inspection data. He used a segment ofan oil and gas pipeline of 914 mm diameter and8.74 mm wall thickness to illustrate his approach. Wewill consider the same case considered by Pandey(1998) to illustrate the approach developed in thepresent work. Input data used in this example aregiven in Table II. Pandey (1998) showed that thetime of inspection should be between 5 and 20 yearsdepending upon the repair criteria (considering corro-sion to be the only degradation mechanism). He alsoconcluded that a single inspection after 10 years witha repair criterion of 1.4 would minimize the pipelinefailure probability to 0.01 per km over 25 years.

7. RESULTS AND DISCUSSIONS

This is a comparatively new segment of the oiland gas pipeline and had only one regular inspection.

As may be seen from Fig. 7 (for thinning), the peaksof the prior and the posterior functions are very muchseparated from each other. The prior is more spread ascompared to the posterior (perfect and imperfect in-spections). The posterior distribution for imperfect in-spections shows the least spread. The results for crackgrowth rate are shown in Fig. 8. In this case the peaksof the posterior functions occur at degradation ratesof 0.2 and 0.195 mm/year for imperfect and prefectinspections, respectively. These peaks are away fromthe prior peak, which occured at a degradation rateof 0.12 mm/year.

A detailed cost analysis is conducted for bothdegradation mechanisms based on the posterior forimperfect inspections. The results are shown in Figs. 9and 10. When corrosion is the dominant degradationmechanism, it was found that the preventive replace-ment time is 14 years and the failure time is 26 years(see Fig. 9). The results considering SCC to be thedominant mechanism are 23 years for the replacementand 62 years for failure (see Fig. 10). It is evident thatthe thinning degradation mechanism is fast and thusmust be used as the basis in inspection planning. Thecost curve has two minima, one at 10 years and theother at 22 years. The first minimum occurring at aninterval of 10 years (though not the global minimum)is a more conservative basis for inspection. Since thepipeline has already been in operation for 5 years,the next inspection is due in 5 years. The measure-ments obtained from the next inspection can then beused to update the distribution and to predict a better

Risk-Based Integrity and Inspection Modeling 217

Table II. Input Details for Segment of

Oil and Gas Pipeline

Background information 5 years of operation

Design temperature 120◦C and pressure 7.0 MPa

Operating temperature 40◦C and pressure 5.7 Barg

Pipe dimensions 914 mm diameter; 1 km length, 8.74 mm wall thickness

Material of construction Steel grade × 60

Active damage mechanism Stress corrosion cracking

Internal thinning

Inspection history One inspection, defect of depth (mean) 0.5 mm, CoV 0.20

Other Parameters ValuesYield strength Mean 461 MPa, coefficient of variance (CoV) 0.20

Operating pressure Mean 5.7 MPa, CoV 0.05

Corrosion rate Mean 0.15 mm/year; CoV 0.25

SCC rate Mean 0.1 mm/year; CoV 0.25

Corrosion allowance 2.5 mm

n (power function for crack 0.50

growth rate)

Crack length-to-depth ratio 6

Inspection cost $5,000

Preventive replacement cost $10,000

Failure cost $100,000

estimate for the following inspection time. If data ob-tained from the next inspection support the currentposterior then the following inspection will coincidewith the preventive replacement of this pipeline. Theoptimum inspection intervals are calculated, based

0

5

10

15

20

25

30

35

40

45

50

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Corrosion rate (mm/year)

De

ns

ity

prior

postprior

post_imp_prior

post_imp_prior_CDF

Fig. 7. Prior and posterior distribution

for corrosion degradation mechanism.

on SCC degradation mechanism, as 20 and 55 years.Because corrosion effects are more dominant in thecase of a pipeline, the SSC-based inspection intervalis not of importance. However, in the next inspection(5 years from now) inspection for crack growth should

218 Khan, Haddara, and Bhattacharya

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2 2.5

Crack growth rate (mm/year)

De

ns

ity

0

0.2

0.4

0.6

0.8

1

1.2

Cu

mu

lati

ve

de

ns

ity

fu

nc

tio

n

prior

postprior

post_imp_prior

post_imp_prior_CDF

Fig. 8. Prior and posterior density

distribution for cracking growth in

pipeline.

also be conducted to verify the posterior and subse-quent analysis.

8. CONCLUSIONS

In this work, we have presented a method for thedetermination of optimal maintenance intervals forprocess components. The method takes into accountthe random nature of material degradation of pro-

0

1000

2000

3000

4000

5000

6000

7000

0 5 10 15 20 25 30

Inspection interval (years)

Avera

ge c

ycle

co

st

($/y

ear)

Average cycle cost

Replacement time

Failure time Fig. 9. Average inspection (cost/year) for

corrosion degradation on pipeline.

cess components. The material degradation rate canbe linear or nonlinear. The method allows the up-dating of the probability density function for the ma-terial degradation, using a Bayesian approach. Thus,the failure rate can be adjusted based on actual mea-surements made during inspections. The updated fail-ure rate obtained reflects the changes in operatingand environmental conditions. The inspection inter-vals obtained are more reliable because they arebased on actual measurements. The method takes into

Risk-Based Integrity and Inspection Modeling 219

0

200

400

600

800

1000

1200

1400

1600

1800

0 10 20 30 40 50 60 70 80 90

Inspection interval (years)

Avera

ge c

ost

per

cycle

($/y

ear)

Average cycle cost

Replacement time

Repair timeFig. 10. Average inspection cost ($/year)

for crack degradation mechanism in

pipeline.

consideration the random error that may be as-sociated with the measurements obtained duringinspection.

The method optimizes maintenance intervalsbased on the risk associated with component failure.The optimization criterion is based on a level of riskthat satisfies the acceptable risk criteria. Using riskas an optimization criterion allows the maintenanceinterval to be a function of both the probability of fail-ure as well as the consequences of this failure. Properattention can then be devoted to the maintenance ofcritical equipment.

The gamma distribution seems to describe thematerial degradation process well. Results obtainedusing the assumption that measurements obtainedduring inspections are imperfect are more realisticthan those assuming perfect measurements.

The method has been applied to two case studies:an autoclave and a pipeline. The results obtained forthese two case studies are in general agreement withresults discussed in the literature. Results of the casestudies presented above show that the method pro-duces reliable estimates for the inspection intervals.The optimal inspection intervals given by the modelfor the case studies considered are reasonable and ingeneral agreement with results obtained using differ-ent approaches.

The main disadvantage of the method is thatit is computationally intensive. For the autoclavecase study, the simulation using 200 samples (codedin MATLAB) took about 8 hours, whereas for the

pipeline case study, it took about 5 hours of computa-tion time on a personal computer with an Intel Pen-tium 4 (clocks at 1.81GHz) and 256 MB RAM.

However, the method can be easily programmedand the data required can be easily obtained fromoperational records and original design documents.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the finan-cial support of Natural Science and Engineering Re-search Council (NSERC) under a discovery grant andCanada Foundation of Innovation (CFI) new oppor-tunity award.

NOMENCLATURE

A Crack depth, mmD Diameter of the component (vessel,

pipe, etc.), mmC Corrosion rate, mm/year

Ccr Crack growth rate, mm/yearcmax Corrosion allowance, mm

ci(ρ, �k) Costs incurred during unit time (i −1, i) as a function of replacementpercentage and inspectional inter-val, $

C(ρ, �k) Expected average costs per unittime as function of replacementpercentage and inspection interval,$/year

220 Khan, Haddara, and Bhattacharya

E(X) Expectation of the random variableX

f X(X) Probability density function of therandom variable X

G State function, dimensionlessGa(x | α, β) Gamma density of X as a function

of x with the shape parameter α andthe scale parameter β

Ig(x | α, β) Inverted gamma density of X as afunction of x with the shape param-eter α and the scale parameter β

KIC Material fracture toughness,MPa(mm)1/2

lcr Characteristic crack length, mmL(μ | x) Likelihood of the variable μ given

the measurement xm Safety margin, the amount of degra-

dation at which the component is as-sumed to fail, dimensionless

N Crack growth exponent; for exam-ple, PWHT components n = 0.25and for non-PWHT components n= 0.50.

P Operating pressure, kPaP(μ | y) Conditional probability of failure

due to wall loss y, given degradationrate μ

Rl/a Crack length-to-depth (thickness)ratio; it is considered as 6

R Material resistance, MPaR(φ, τ ) Risk factor, dollar per unit time

E|R(φ, τ )| Expected average risk factor, dollarper unit

S Material strength, MPaSt Applied stress, MPa

thickness Thickness of the material of con-struction, mm

X(t) Random variable in stochastic pro-cess

Y(t) Variable in stochastic process con-sisting of variable X(t) and an error,X(t) + ε

Ye Geometric factor, dimensionless

GREEK SYMBOLS

α Shape parameter for gamma and in-verted gamma densities

β Scale parameter for gamma and in-verted gamma densities

E Normally distribution error with mean0 and standard deviation σ ε

δk Difference of error between two suc-cessive inspections, εk and εk−1

μ Mean of the uncertain gamma density(random variable X)

σ Standard deviation of the random vari-able X

υ Coefficient of varianceρ Replacement stage, percentage of

safety margin at which the componentshould be replaced

φ{m, ρ} Ratio of replacement to failure leveland calculated for each time interval τ

τ Time interval� (a) Gamma function

�(a, x) Incomplete gamma functionπ(μ) Prior density of the variable μ

�tk Time interval between two inspections

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